Jacek Szmigielski

Multi-peakons and a theorem of Stieltjes

A closed form of the multi-peakon solutions of the Camassa-Holm equation is found using a theorem of Stieltjes on continued fractions. An explicit formula is obtained for the scattering shifts.

Multi-peakon solutions of the Degasperis–Procesi equation

We present an inverse scattering approach for computing n-peakon solutions of the Degasperis-Procesi equation (a modification of the Camassa-Holm (CH) shallow water equation). The associated non-self-adjoint spectral problem is shown to be amenable to analysis using the isospectral deformations induced from the n-peakon solution, and the inverse problem is solved by a method generalizing the continued fraction solution of the peakon sector of the CH equation.

Degasperis-Procesi peakons and the discrete cubic string

We use an inverse scattering approach to study multi-peakon solutions of the Degasperis–Procesi (DP) equation, an integrable PDE similar to the Camassa–Holm shallow water equation. The spectral problem associated to the DP equation is equivalent under a change of variables to what we call the cubic string problem, which is a third order non-selfadjoint generalization of the well-known equation describing the vibrational modes of an inhomogeneous string attached at its ends. We give two proofs that the eigenvalues of the cubic string are positive and simple; one using scattering properties of DP peakons, and another using the Gantmacher–Krein theory of oscillatory kernels. For the discrete cubic string (analogous to a string consisting of n point masses) we solve explicitly the inverse spectral problem of reconstructing the mass distribution from suitable spectral data, and this leads to explicit formulas for the general n-peakon solution of the DP equation. Central to our study of the inverse problem is a peculiar type of simultaneous rational approximation of the two Weyl functions of the cubic

The Cauchy Two-Matrix Model

We introduce a new class of two(multi)-matrix models of positive Hermitian matrices coupled in a chain; the coupling is related to the Cauchy kernel and differs from the exponential coupling more commonly used in similar models. The correlation functions are expressed entirely in terms of certain biorthogonal polynomials and solutions of appropriate Riemann–Hilbert problems, thus paving the way to a steepest descent analysis and universality results. The interpretation of the formal expansion of the partition function in terms of multicolored ribbon-graphs is provided and a connection to the O(1) model. A steepest descent analysis of the partition function reveals that the model is related to a trigonal curve (three-sheeted covering of the plane) much in the same way as the Hermitian matrix model is related to a hyperelliptic curve.

Peakon-Antipeakon Interaction

Abstract Explicit formulas are given for the multi-peakon-antipeakon solutions of the Camassa– Holm equation, and a detailed analysis is made of both short-term and long-term aspects of the interaction between a single peakon and single anti-peakon.

An Inverse Spectral Problem Related to the Geng–Xue Two-Component Peakon Equation

We solve a spectral and an inverse spectral problem arising in the computation of peakon solutions to the two-component PDE derived by Geng and Xue as a generalization of the Novikov and Degasperis ...

Peakons, strings, and the finite Toda lattice

As is well-known, the Toda lattice flow may be realized as an isospectral flow of a Jacobi matrix. A bijective map from a discrete string problem with positive weights to Jacobi matrices allows the pure peakon flow of the Camassa-Holm equation to be realized as an isospectral Jacobi flow as well. This gives a unified picture of the Toda, Jacobi, and multipeakon flows, and leads to explicit solutions of the Jacobi flows via Stieltjes’ determination of the continued fraction expansion of a Stieltjes transform. A simple modification produces a bijection from generalized strings, with positive and negative weights, to singular Jacobi matrices, and thus brings peakon/antipeakon flows into the same picture. c

Cauchy biorthogonal polynomials

The paper investigates the properties of certain biorthogonal polynomials appearing in a specific simultaneous Hermite-Pade approximation scheme. Associated with any totally positive kernel and a pair of positive measures on the positive axis we define biorthogonal polynomials and prove that their zeros are simple and positive. We then specialize the kernel to the Cauchy kernel 1x+y and show that the ensuing biorthogonal polynomials solve a four-term recurrence relation, have relevant Christoffel-Darboux generalized formulas, and their zeros are interlaced. In addition, these polynomials solve a combination of Hermite-Pade approximation problems to a Nikishin system of order 2. The motivation arises from two distant areas; on the one hand, in the study of the inverse spectral problem for the peakon solution of the Degasperis-Procesi equation; on the other hand, from a random matrix model involving two positive definite random Hermitian matrices. Finally, we show how to characterize these polynomials in terms of a Riemann-Hilbert problem.

The string density problem and the Camassa-Holm equation

Recently, the string density problem, considered in the pioneering work of M. G. Krein, has arisen naturally in connection with the Camassa–Holm equation for shallow water waves. In this paper we review the forward and inverse string density problems, with some numerical examples, and relate it to the Camassa–Holm equation, with special reference to multi-peakon/anti-peakon solutions. Under stronger assumptions, the Camassa–Holm spectral problem and the string density problem can be transformed to the Schrödinger spectral problem and its inverse problem. Recent results exploiting this transformation are reviewed briefly.

Systems of PDEs Obtained from Factorization in Loop Groups

AbstractWe propose a generalization of a Drinfeld–Sokolov scheme of attaching integrable systems of PDEs to affine Kac–Moody algebras. With every affine Kac–Moody algebra